8. Encapsulation

In this section you are going to learn how to put a nice little knot in a box, what I call "encapsulation". It is based on a mathematical notion called "duality", but stop! don’t run away, it’s easier to begin with an example to show you that 1. it’s easy; 2. you really need it if you want to do anything big in the celtic knot business.

Suppose you like the trefoil knot (I do!) and you want to do something with it. For example, a cross. As the trefoil is coded by a triangle, you try and put four triangles together. After a lot of sweat, curses and errors, you end up with something like that:

That’s very nice but requires a lot of fine hand tuning. You can try and go further and make a bigger cross with five of these little crosses. After a lot of bad words, you may get something like this:

Very nice indeed but difficult to adjust isn’t it? If you tried, you realized you cannot do large and precise figures this way. Fortunately, there is another way, it’s called encapsulation. But first you have to learn things about a mathematical (don’t run!) notion:

Duality

Going back to the trefoil, suppose, when you extract its graph, you don’t put the vertices into the white but into the black regions (see Extraction). Of course, the infinite region has to be split into several vertices not just one and these vertices are simbolically fused together by a continuous wall surrounding the whole knot. In pictures rather than words, you would end up with figure c, not figure b:

What you’ve just witnessed is the construction of the dual graph of the triangle. Once you have the graph, you don’t need the knot, you can construct its dual straight away: The vertices of the dual graph are in the middle of the faces of the original graph and its edges are transverse to each edge:

For example the triangular and hexagonal lattices are dual to each other. The dual of one triangle is a portion of the hexagonal lattice.

Now you see that it’s easier to form a little cross using the dual graph, the knot is completley inside a box, a triangular capsule! You just draw a square, add its two diagonals to make four triangles, copy the dual motif inside each triangle and open some edges in the walls so that the trefoils melt into one. In pictures:

And it’s even easier to construct the bigger cross, simply stack five square such as that one and open some doors to let them escape from their boxes:

So the philosophy is that to make a big knotwork, you first have to find small knots you like, work out their dual graph which are enclosed in a capsule and use these boxes to tile the area you want to design. Let’s have a look at another example. Playing with triangular lattices, you stumbled onto that nice motif:

To make something with it, you first have to work out its dual graph:

Then you can make for example a five branched star:

Now if you want to do knots which have a beginning and an end, you have to switch to the older version of the site and look for entanglements.

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Forum messages

I don’t understand the dual graph. In your drawing of the trefoil knot, you ignore a lot of lines in the dual graph. What’s the purpose of the dual graph if you don’t follow it? And how do you find the dual graph without the original knot? <: (

C. Coire

NB: your flashing pictures are distracting and difficult to understand.

> In your drawing of the trefoil knot, you ignore a lot of lines in the dual graph.

I don’t. The one outside are walls, they can not be crossed. The thread
"bounces back" inside the triangle.

> What’s the purpose of the dual graph if you don’t follow it? And how do you find the dual graph without the original knot? <: (

You have to begin somewhere. If you have a graph, the way to build the
dual graph is to put a dual vertex "in the middle" of each face. Each
edge separates 2 faces, which are associated with 2 dual vertices. Draw
a dual edge between these 2 vertices, over and across the original edge.

Hi. I had some experiences with dual maps to. It also appear in our electronics class. The idea (to me) is this: replace faces/area with points (in our class we call it meshes and nodes: faces -> meshes, points -> nodes) and points with faces. That area outside is also considered and in its dual its a point. This tutorial illustrated it as a very thick line, in our class we just represent it with a point and draw long lines to connect it to other parts.

Now when to faces share an line/edge, its dual are points connected together by a line/edge. When two points are connected together by a line its dual are two faces sharing a line/edge.

example: suppose you have a triangle. A triangle is a figure that has:

three points (corners) connected to the other by a line
two faces ( the inside one and the outside one ) that share three edges.

So its dual is a figure that has:

three faces that share one edge with the other faces
two points that are connected by three lines

The closest description of the triangle’s dual figure that I could give is a three sided banana. The sides of the banana are the three faces. The tips of the banana are the two points. Could you see it?

Another. Say a circle:

two faces that share one edge.
a point (virtual) that is connected by a line to itself

Its dual:

two points connected by a line
a face that share an edge to itself

If you can’t visualize its dual. I tell you its a 2d line segment. A line segment has two points connected by a line and a "face" (the pane it is on) that share an edge with itself.

Hahaha. Isn’t it funny? I think it is clever.

This branch of specialized geometry is called topology. It has found very important applications where the data can be represented by lines and points. Example: electronics, lines are the elements (resistor, dry cell, LED) and the points are the wires connecting them; traffic, lines are the roads and points are the intersections; computer networks, lines are the cables, fiber optics, etc. and the points are the computers; and many more.

I don’t understand how to make a duel diagram into a knot. I finally figured out how to do it with the original diagrams, after much studying of the figures and practice.
Can you explain how to make the duels into knots?

Once you get it into a graph, the dual and the primal behave exactly the same: put a cross at the middle of each edge; to connect the threads follow the wall, turn at the corner, follow the wall; to decide the over/under, use the guide...

Everything is the same. Except that the dual is boxed in walls, that’s the beauty of it. That means that boundary edges are walls, don’t put crossings there.

Dear B. Turneabe, indeed, you don’t need the primal graph, once you’ve got the dual, you can go without the primal, just follow the same rules. On the other hand, both graphs at the same time help you to layout neatly your threads, especially when you try to make them plump and fat: you have to turn around both corners, around the primal vertices and around the dual one; it constrains a lot your string and it is good so.

I asked because I am having difficulty drawing the duel from the primal. I seem to be able to draw both the primal and the duel from the knot, independently, without much trouble. But primal TO duel, trouble. I have no idea why.

I know you don’t have much time to spare, but could you post a few extra examples? and exercises? Practice makes perfect!

You don’t actually need to be a skilled drawer to do without the primary: it is exactly the same method (follow the wall, turn the corner, follow the wall, connect), whether you begin with the primary or with the dual graph. The only difference is that the dual graph is enclosed in a boundary of walls whereas the primary flows more freely around the corners.

But remember that if you want to design elaborate large knotwork, you’ll have to put several motives next to one another, and to achieve that, you have to be able to go from one primal graph to its dual. But alright, once you have your gallery of dual graphs, you are set and you don’t need their primal any longer.